The Davies Quantile Function.

The Davies distribution

Overview

The Davies distribution is a flexible family of distributions for non-negative observations; it is particularly suitable for right-skewed data. Hankin and Lee (2006) set out mathematical properties of the Davies distribution and the Davies package is showcased here. It is defined in terms of its quantile function

We may sample from this distribution using rdavies():

params <- c(2,0.1,0.1)
rdavies(10,params)
#>  [1] 1.761097 2.008966 1.767981 2.020754 1.674392 2.003635 1.485477 1.980971
#>  [9] 2.253223 2.567022

Moments are given by where is the beta function. In the package this is given by M(), which is a convenience wraper for davies.moment(). Numerical verification for the second (non-central) moment:

c(mean(rdavies(1e6,params)^2),M(2,params))
#> [1] 4.273915 4.275837

Estimation

The least-squares technique described in Hankin and Lee 2006 is not implemented, but the package implements a maximum-likelihood estimate:

x <- rdavies(80,params)
p_estimate <- maximum.likelihood(x)
p_true <- params
p_estimate
#> [1] 1.95306332 0.08418046 0.11483549
(bias <- p_estimate - p_true)
#> [1] -0.04693668 -0.01581954  0.01483549

Reference

Robin K. S. Hankin and Alan Lee 2006. “A new family of non-negative distributions”. Aust. N. Z. J. Stat, 48(1):67-78